A theoretical basis of the moment magnitude scale

This is the authors' abstract. We don't add key points for this paper.

To measure the size of an earthquake, we can use two physically different quantities: the radiated seismic energy E_R E R , from which seismic magnitude scales are defined, and the seismic moment M₀ M 0 . The former depends on the dynamic process of earthquake faulting, whereas the latter does not. Nevertheless, if there exists a scaling relation between E_R E R and M₀ M 0 , we can define a magnitude scale derived from the seismic moment. The widely accepted moment magnitude scale, M_w = (log M₀ - 9.1) / (log M₀ - 9.1) 1.5 . -0pt 1.5 M w = ( log M 0 - 9.1 ) ( log M 0 - 9.1 ) 1.5 1.5 in MKS units, was established by substituting the scaling relation of E_R = 5 × 10- 5 M₀ E R = 5 × 10 - 5 M 0 into the Gutenberg-Richter empirical energy–magnitude relation, log E_R = 1.5M_s + 4.8 log E R = 1.5 M s + 4.8 . The above scaling relation comes from the energy–moment relation, E_R = (Δ τ / Δ τ 2μ . -0pt 2μ )M₀ E R = ( Δ τ Δ τ 2 μ 2 μ ) M 0 , based on a simplified energy balance equation in earthquake faulting with a uniform stress drop Δ τ Δ τ . However, this energy–moment relation is a bit strange, because the right-hand side depends exclusively on the final static state of earthquake faulting, while the left-hand side must depend on the whole dynamic process of earthquake faulting. Theoretically, the radiated seismic energy E_R E R is expressed as a function of cumulative moment <jats:a